Turn Skipping and the Game Coloring Number

The game coloring number gcol($G$) of a graph $G$ is a two player competitive variant of the coloring number. We introduce the preordered game coloring number to study the consequences of either player skipping any number of turns. In particular, we show that neither player can improve their performance by doing so. We use this result to show that for any induced subgraph $H \subset G$, $|G-H| = k$ implies the tight bound gcol($H$) $\leq$ gcol($G$) $-$ $2k$ and if $d_G(x)$ $\leq$ gcol($G-x$), then gcol($G-x$) $\leq$ gcol($G$) $-$ $1$.